We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N = 2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2, Z) Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.
Dyonic 1/4-BPS states in Type IIB string theory compactified on K3 × T 2 are counted by meromorphic Jacobi forms. The finite parts of these functions, which are mixed mock Jacobi forms, account for the degeneracy of states stable throughout the moduli space of the compactification. In this paper, we obtain an exact asymptotic expansion for their Fourier coefficients, refining the Hardy-Ramanujan-Littlewood circle method to deal with their mixed-mock character. The result is compared to a low-energy supergravity computation of the exact entropy of extremal dyonic 1/4-BPS single-centered black holes, obtained by applying supersymmetric localization techniques to the quantum entropy function.
Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3 elliptic genera twined by discrete symmetries that are manifest in the UV description, though often inaccessible in the IR. We obtain various twining functions coinciding with moonshine predictions that have not been observed in physical theories before. These include twining functions arising from Mathieu moonshine, other cases of umbral moonshine, and Conway moonshine. For instance, all functions arising from M 11 ⊂ 2.M 12 moonshine appear as explicit twining genera in the LG models, which moreover admit a uniform description in terms of its natural 12-dimensional representation. Our results provide strong evidence for the relevance of umbral moonshine for K3 symmetries, as well as new hints for its eventual explanation.
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